the $ pq $ analogue to Serre's conjecture (see "Mod pq Galois representations and Serre & # 39; s Conjecture" – Khare, Kiming) says that if $ bar { rho} _1: G _ { mathbb {Q}} rightarrow text {GL} _2 ( mathbb {F} _p) $ It is a mod $ p $ Representation of Galois and $ bar { rho_2}: G _ { mathbb {Q}} rightarrow text {GL} _2 ( mathbb {F} _q) $ It is a mod $ q $ Representation of Galois, then yes $ bar { rho} _1 $ Y $ bar { rho} _2 $ they are odd, irreducible, branched into only a finite number of prime numbers and of the same "weight", then under some additional hypotheses, $ bar { rho} _1 $ Y $ bar { rho} _2 $ Both elevate to Galois representations coming from the same own form.

Before Serre made his conjecture (for example, a cousin), there was certainly a large amount of computational evidence, is there any computational evidence for this conjecture?

It seems that the easiest way to find examples of $ bar { rho} _1 $ Y $ bar { rho} _2 $ as described would be to take the mod $ p $ and mod $ q $ representations associated with a new form and one would have to build to find $ bar { rho} _1 $ Y $ bar { rho} _2 $ That do not arise in this way to find computational evidence.